Insecurity for compact surfaces of positive genus

Abstract

A pair of points in a riemannian manifold $M$ is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in $M$ are secure. A manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. We prove this for surfaces of genus greater than zero. We also prove that a closed surface of genus greater than one with any riemannian metric and a closed surface of genus one with generic metric are totally insecure.Comment: 37 pages, 11 figure

Real analytic metrics on S 2 with total absence of finite blocking

“…The main result of this paper is the following: or P 2 (R), since totally insecure metrics for positive genus surfaces were already obtained in [3], as described in our introduction.…”

“…The real analytic metrics h on S 2 for which we prove total insecurity are obtained in the same way as the metrics on S 2 for which K. Burns and Gerber [8,9] showed that the geodesic flow is ergodic. Our proof relies on ideas in [3] and techniques in non-uniform hyperbolicity [4,18]. Burns and Gutkin [6] and, independently, J.-F. Lafont and B. Schmidt [19] showed that compact Riemannian manifolds (of any dimension) with no conjugate points whose geodesic flows have positive topological entropy are totally insecure.…”

“…(See Proposition 6.9 for a precise statement.) This condition is essentially taken from [3], and it is easy to establish if (S 2 , h) is replaced by a manifold of negative curvature and ρ is replaced by any closed geodesic. (For manifolds of negative curvature, the first statement in Proposition 6.6 is also easy to prove, for any closed geodesic ρ, but the analog of Proposition 6.9 can be proved directly.…”

Locally Lipschitz graph property for lines

“…They were widely studied and applied in the literature, for example, [3], [4], [5], [6], [7], [10], [15], [18], [20], [31]. They were widely studied and applied in the literature, for example, [3], [4], [5], [6], [7], [10], [15], [18], [20], [31].…”

A dense G-delta set of Riemannian metrics without the finite blocking property

“…V. Bangert and E. Gutkin obtained stronger results for the case when the dimension of M is two and the genus is positive [1]. They proved that if M has genus greater than one, then every Riemannian metric is totally insecure.…”